Next, for given initial conditions w⁡(0)=0 and w′⁡(0)=k, with k real,
w⁡(x) has at least one pole on the real axis. There are two special values of
k, k1 and k2, with the properties -0.45142 8<k1<-0.45142 7,
1.85185 3<k2<1.85185 5, and such that:

(a)

If k<k1, then w⁡(x)>0 for x0<x<0, where x0 is the first
pole on the negative real axis.

(b)

If k1<k<k2, then w⁡(x) oscillates about, and is asymptotic to,
-16⁢|x| as x→-∞.

(c)

If k2<k, then w⁡(x) changes sign once, from positive to negative,
as x passes from x0 to 0.

For illustration see Figures 32.3.1 to
32.3.4, and for further information see
Joshi and Kitaev (2005),
Joshi and Kruskal (1992), Kapaev (1988), Kapaev and Kitaev (1993), and
Kitaev (1994).

Any nontrivial real solution of (32.11.4) that satisfies
(32.11.5) is asymptotic to k⁢Ai⁡(x), for some nonzero
real k, where Ai denotes the Airy function (§9.2).
Conversely, for any nonzero real k, there is a unique solution wk⁡(x)
of (32.11.4) that is asymptotic to k⁢Ai⁡(x) as
x→+∞.

If |k|<1, then wk⁡(x) exists for all sufficiently large |x| as
x→-∞, and

For illustration see Figures 32.3.5 and
32.3.6, and for further information see
Ablowitz and Clarkson (1991), Bassom et al. (1998), Clarkson and McLeod (1988),
Deift and Zhou (1995), Segur and Ablowitz (1981), and
Suleĭmanov (1987). For numerical studies see Miles (1978, 1980) and Rosales (1978).

Any nontrivial solution of (32.11.29) that satisfies
(32.11.30) is asymptotic to
h⁢U2⁡(-ν-12,2⁢x) as x→+∞, where h(≠0) is a constant. Conversely, for any h(≠0) there is a unique
solution wh⁡(x) of (32.11.29) that is asymptotic to
h⁢U2⁡(-ν-12,2⁢x) as x→+∞. Here
U denotes the parabolic cylinder function (§12.2).